The Pound–Drever–Hall (PDH) technique is a widely used and powerful approach for stabilizing the frequency of
light
Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
emitted by a
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
by means of
locking to a stable cavity. The PDH technique has a broad range of applications including
interferometric gravitational wave detectors,
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, and
time measurement standards, many of which also use related techniques such as
frequency modulation
Frequency modulation (FM) is a signal modulation technique used in electronic communication, originally for transmitting messages with a radio wave. In frequency modulation a carrier wave is varied in its instantaneous frequency in proporti ...
spectroscopy. Named after
R. V. Pound,
Ronald Drever
Ronald William Prest Drever (26 October 1931 – 7 March 2017) was a Scottish experimental physicist. He was a professor emeritus at the California Institute of Technology, co-founded the LIGO project, and was a co-inventor of the Pound–Drever ...
, and
John L. Hall, the PDH technique was described in 1983 by Drever, Hall and others working at the
University of Glasgow
The University of Glasgow (abbreviated as ''Glas.'' in Post-nominal letters, post-nominals; ) is a Public university, public research university in Glasgow, Scotland. Founded by papal bull in , it is the List of oldest universities in continuous ...
and the U. S.
National Bureau of Standards
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sc ...
.
This optical technique has many similarities to an older frequency-modulation technique developed by Pound for microwave cavities.
[ (Pedagogical review article describing the technique.)]
Since a wide range of conditions contribute to determine the
linewidth
A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used ...
produced by a laser, the PDH technique provides a means to
control and decrease the laser's linewidth, provided an
optical cavity
An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that confines light waves similarly to how a cavity resonator confines microwaves. Optical cavities are a major component of lasers, ...
that is more stable than the laser source. Alternatively, if a stable laser is available, the PDH technique can be used to stabilize and/or measure the instabilities in an optical cavity length.
The PDH technique responds to the frequency of laser emission independently of intensity, which is significant because many other methods that control laser frequency, such as a side-of-fringe lock are also affected by intensity instabilities.
Laser stabilization
In recent years the Pound–Drever–Hall technique has become a mainstay of laser frequency stabilization. Frequency stabilization is needed for high precision because all lasers demonstrate frequency wander at some level. This instability is primarily due to temperature variations, mechanical imperfections, and laser gain dynamics,
which change laser cavity lengths, laser driver current and voltage fluctuations, atomic transition widths, and many other factors. PDH locking offers one possible solution to this problem by
actively tuning the laser to match the resonance condition of a stable reference cavity.
The ultimate linewidth obtained from PDH stabilization depends on a number of factors. From a signal analysis perspective, the noise on the locking signal can not be any lower than that posed by the
shot noise
Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process.
In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where s ...
limit.
However, this constraint dictates how closely the laser can be made to follow the cavity. For tight locking conditions, the linewidth depends on the absolute stability of the cavity, which can reach the limits imposed by thermal noise.
Using the PDH technique, optical linewidths below 40 mHz have been demonstrated.
Applications
Prominently, the field of
interferometric
Interferometry is a technique which uses the '' interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber opt ...
gravitational wave
Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...
detection depends critically on enhanced sensitivity afforded by optical cavities.
The PDH technique is also used when narrow spectroscopic probes of individual quantum states are required, such as
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
,
time measurement standards, and
quantum computer
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of both particles and waves, and quantum computing takes advantage of this behavior using specialized hardware. ...
s.
Overview of technique
Phase modulated light, consisting of a carrier frequency and two side bands, is directed onto a two-mirror cavity. Light reflected off the cavity is measured using a high speed
photodetector
Photodetectors, also called photosensors, are devices that detect light or other forms of electromagnetic radiation and convert it into an electrical signal. They are essential in a wide range of applications, from digital imaging and optical ...
; the reflected signal consists of the two unaltered side bands along with a phase-shifted carrier component. The photodetector signal is
mixed down with a
local oscillator
In electronics, the term local oscillator (LO) refers to an electronic oscillator when used in conjunction with a Frequency mixer, mixer to change the frequency of a signal. This frequency conversion process, also called Heterodyne, heterodyning ...
, which is in phase with the light modulation. After phase shifting and
filtering
Filtration is a physical process that separates solid matter and fluid from a mixture.
Filter, filtering, filters or filtration may also refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Fil ...
, the resulting electronic signal gives a measure of how far the laser carrier is off resonance with the cavity and may be used as feedback for active stabilization. The feedback is typically carried out using a
PID controller
PID or Pid may refer to:
Medicine
* Pelvic inflammatory disease or pelvic inflammatory disorder, an infection of the upper part of the female reproductive system
* Primary immune deficiency, disorders in which part of the body's immune system is ...
which takes the PDH error signal readout and converts it into a voltage that can be fed back to the laser to keep it locked on resonance with the cavity.
The main innovation of the PDH technique is to monitor the ''derivative'' of the cavity transmission with respect to detuning, rather than the cavity transmission itself, which is symmetric about the resonant frequency. Unlike a side-of-fringe lock, this allows the sign of the feedback signal to be correctly determined on both sides of resonance. The derivative is measured via rapid modulation of the input signal and subsequent mixing with the drive waveform, much as in
electron paramagnetic resonance
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spin ...
.
PDH readout function
The PDH readout function gives a measure of the resonance condition of a cavity. By taking the derivative of the cavity
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
(which is symmetric and
even) with respect to frequency, it is an
odd function of frequency and hence indicates not only whether there is a mismatch between the output frequency ''ω'' of the laser and the resonant frequency ''ω''
res of the cavity, but also whether ''ω'' is greater or less than ''ω''
res. The
zero-crossing of the readout function is sensitive only to intensity fluctuations due to the frequency of light in the cavity and insensitive to intensity fluctuations from the laser itself.
Light of
frequency
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
can be represented mathematically by its electric field, ''E''
0''e
iωt''. If this light is then phase-modulated by ''β''sin(''ω''
m''t''), where ''ω''
m is the modulation frequency and ''β'' is the modulation depth, the resulting field ''E''
i is
:
This field may be regarded as the
superposition
In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
of three frequency components. The first component is an electric field of angular frequency ''ω'', known as the ''carrier'', and the second and third components are fields of angular frequency and , respectively, called the ''
sideband
In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency, that are the result of the modulation process. The sidebands carry the information transmitted by the radio signal. The sidebands c ...
s''.
In general, the light ''E''
r reflected out of a
Fabry–Pérot two-mirror cavity is related to the light ''E''
i incident on the cavity by the following
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
:
:
where , and where ''r''
1 and ''r''
2 are the
reflection coefficients of mirrors 1 and 2 of the cavity, and ''t''
1 and ''t''
2 are the
transmission coefficients of the mirrors.

Applying this transfer function to the phase-modulated light ''E''
i gives the reflected light ''E''
r:
[The transfer function ''R'' is applied independently to each of the three exponential terms because a Fabry–Perot cavity is a ]linear time-invariant system
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of Linear system#Definition, linearity and Time-invariant system, ...
. The cavity's response to light of frequency ''ω''1 is the same regardless of whether it is also simultaneously responding to light of some other frequency ''ω''2.
:
The power ''P
r'' of the reflected light is proportional to the square magnitude of the electric field, ''E
r* E
r'', which after some algebraic manipulation can be shown to be
:
Here ''P''
0 ∝ , ''E''
0,
2 is the power of the light incident on the Fabry–Pérot cavity, and ''χ'' is defined by
:
This ''χ'' is the ultimate quantity of interest; it is an antisymmetric function of . It can be extracted from ''P''
r by
demodulation
Demodulation is the process of extracting the original information-bearing signal from a carrier wave. A demodulator is an electronic circuit (or computer program in a software-defined radio) that is used to recover the information content fro ...
. First, the reflected beam is directed onto a
photodiode
A photodiode is a semiconductor diode sensitive to photon radiation, such as visible light, infrared or ultraviolet radiation, X-rays and gamma rays. It produces an electrical current when it absorbs photons. This can be used for detection and me ...
, which produces a voltage ''V''
r that is proportional to ''P''
r. Next, this voltage is
mixed with a phase-delayed version of the original modulation voltage to produce
:
:
Finally,
is sent through a
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
to remove any sinusoidally oscillating terms. This combination of mixing and low-pass filtering produces a voltage ''V'' that contains only the terms involving ''χ'':
:
In theory, ''χ'' can be completely extracted by setting up two demodulation paths, one with and another with . In practice, by judicious choice of ''ω''
m it is possible to make ''χ'' almost entirely real or almost entirely imaginary, so that only one demodulation path is necessary. ''V''(''ω''), with appropriately chosen ''φ'', is the PDH readout signal.
Notes
References
{{DEFAULTSORT:Pound-Drever-Hall technique
Synchronization
Optical devices